{"paper":{"title":"Group actions and a multi-parameter Falconer distance problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Alex Rice, Kyle Hambrook","submitted_at":"2017-05-10T17:46:44Z","abstract_excerpt":"In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\\textbf{d}}=(d_1,d_2, \\dots, d_{\\ell})\\in \\mathbb{N}^{\\ell}$ with $d_1+d_2+\\dots+d_{\\ell}=d$ and $E \\subseteq \\mathbb{R}^d$, we define $$ \\Delta_{{\\textbf{d}}}(E) = \\left\\{ \\left(|x^{(1)}-y^{(1)}|,\\ldots,|x^{(\\ell)}-y^{(\\ell)}|\\right) : x,y \\in E \\right\\} \\subseteq \\mathbb{R}^{\\ell}, $$ where for $x\\in \\mathbb{R}^d$ we write $x=\\left( x^{(1)},\\dots, x^{(\\ell)} \\right)$ with $x^{(i)} \\in \\mathbb{R}^{d_i}$.\n  We ask how large does the Hausdorff dimension of $E$ need to be to ensure t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.03871","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}